Square Root of 8200

The square root is the inverse of the square of the number. 8200 is not a perfect square. The square root of 8200 is expressed in both radical and exponential form. In the radical form, it is expressed as √8200, whereas (8200)^(1/2) in the exponential form. √8200 ≈ 90.5538, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 8200 is broken down into its prime factors.

Step 1: Finding the prime factors of 8200 Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 41: 2^3 x 5^2 x 41

Step 2: Now we found out the prime factors of 8200. The second step is to make pairs of those prime factors. Since 8200 is not a perfect square, the digits of the number can’t be grouped in pairs completely. Therefore, calculating 8200 using prime factorization alone is not sufficient.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 8200, we need to group it as 82 and 00.

Step 2: Now we need to find n whose square is ≤ 82. We can say n as ‘9’ because 9 x 9 = 81, which is lesser than 82. Now the quotient is 9; after subtracting 81 from 82, the remainder is 1.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number 9 + 9 we get 18, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 18n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 18n × n ≤ 100. Let us consider n as 5, now 185 x 5 = 925.

Step 6: Subtract 1000 from 925; the difference is 75, and the quotient is 90.5.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 7500.

Step 8: Now we need to find the new divisor that is 905 because 9055 x 5 = 45275.

Step 9: Subtracting 45275 from 7500, we get the result 2975.

Step 10: Now the quotient is 90.55.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √8200 is approximately 90.55.

The approximation method is another method for finding the square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 8200 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √8200. The smallest perfect square less than 8200 is 8100 and the largest perfect square greater than 8200 is 8281. √8200 falls somewhere between 90 and 91.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (8200 - 8100) ÷ (8281 - 8100) ≈ 0.553. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 90 + 0.553 = 90.553, so the square root of 8200 is approximately 90.553.

• Square root: A square root is the inverse of a square. Example: 5^2 = 25 and the inverse of the square is the square root, that is √25 = 5.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Long division method: A method used to find the square root of a number by dividing it into pairs of digits and performing a series of divisions.
   
• Decimal: A number that includes a whole number and a fractional part separated by a decimal point, for example, 7.86, 8.65, and 9.42.